The waves actually come from a distant star but for our calculations we
assume the waves pass the focal plane as parallel wave fronts. We know that each
point along the parallel wave front must travel an equal distance to the focal
point so line_b + line_c must equal a constant. If we first look at the ray that
is traveling down the mirror axis we see it must travel twice the focal length
to go from a wave front at the focal plane to the mirror and then return to the focus. So b + c
= 2f.
If we want c we can subtract b from both sides of this equation,
c = 2f - b.

We will vary r to find the surface of the parabola. Lets begin by finding b
as a function of r. Using the Pythagorean Theorem, c 2=
r 2+ b 2 . Subtracting r 2
from both sides, b 2 = c
2- r
2.
We know that c = 2f - b so lets try a substitution in order to remove ‘unknown’
c from the formula: